---
tag: D33JK3AXRZ
kind: example
name: Bernoulli scheme
msc: [37A05, 37A35, 37B10]
instance_of: [556WBDAHNP]
uses: [0CQQG3DEW9, 2HWVFHM13R, S3RJQ9CX2P, YRHJA7C7HS, X1D1C7VH6P]
sources:
  - url: https://en.wikipedia.org/wiki/Bernoulli_scheme
same_as:
  wikidata: Q3042439
  wikipedia: Bernoulli scheme
---

## Example

Let $A = \set{1, \dots, k}$ be a finite alphabet and let $p = (p_1, \dots, p_k)$ be a probability vector, with $p_i \ge 0$ and $p_1 + \cdots + p_k = 1$.
Let $X = A^{\Z}$ be the set of two-sided sequences $x = (x_n)_{n \in \Z}$, with the σ-algebra generated by the cylinder sets
$$[a_m, \dots, a_n]_m = \set{x \in X : x_i = a_i \text{ for } m \le i \le n}.$$
Let $\mu$ be the [[infinite product measure|2HWVFHM13R]] of copies of $p$, so $\mu([a_m, \dots, a_n]_m) = p_{a_m} p_{a_{m+1}} \cdots p_{a_n}$.
The shift $\sigma \colon X \to X$, $(\sigma x)_n = x_{n+1}$, is the *Bernoulli scheme* or *Bernoulli shift* $B(p_1, \dots, p_k)$.

Check that $\sigma$ is a [[measure-preserving transformation|556WBDAHNP]]: $x \in \sigma^{-1}[a_m, \dots, a_n]_m$ means $x_{i+1} = a_i$ for $m \le i \le n$.
So $\sigma^{-1}[a_m, \dots, a_n]_m = [a_m, \dots, a_n]_{m+1}$, which has the same measure $p_{a_m} \cdots p_{a_n}$.
Cylinder sets form a π-system generating the σ-algebra, so $\mu(\sigma^{-1} E) = \mu(E)$ for every measurable $E$.

The coordinate functions $x \mapsto x_n$ form an i.i.d. sequence with law $p$, a two-sided version of the [[Bernoulli process|0CQQG3DEW9]] when $k = 2$.
The shift is invertible, [[mixing|S3RJQ9CX2P]], and a [[Kolmogorov automorphism|YRHJA7C7HS]].
Its [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is $-\sum_{i=1}^{k} p_i \log p_i$, computed with the generating partition into the sets $\set{x : x_0 = i}$.
The one-sided version uses $A^{\N}$ instead of $A^{\Z}$ and gives a non-invertible measure-preserving map.

## Explanations

### L1 human

A Bernoulli scheme is the model of rolling the same loaded die once every second, forever into the past and into the future.
A point of the space is a full record of all the rolls, and the shift moves the clock forward by one second.
Because the rolls are independent and identically distributed, moving the clock does not change the probabilities, so the shift preserves the measure.
Bernoulli schemes are the most random measure-preserving systems and serve as the yardstick for the others.